Why Do We Have Functions?

Date: 04/01/2003 at 12:42:43
From: Doctor Peterson
Subject: Re: Functions
Hi, Lisa.
Basically, the concept of functions gives us a way to name the whole
process of evaluating a particular expression, so we can talk about
it as a whole. We can compare different functions, discuss their
properties, or actually operate on functions to make new functions.
It also broadens the concept, because not all functions can be
written as a simple expression. These two processes, naming things
and extending them, are central to what mathematics is all about.
For example, the first function you showed can be called 'squaring',
and the second can be called 'adding 3'; but most functions would
have to have much more complicated names. By calling one F and the
other G, we have a simple way to discuss them. Some functions, like
the square root and the absolute value, can't be expressed in terms
of more basic functions, but only by inventing a whole new symbol. In
fact, we like to write the square root as 'sqrt(x)', using function
notation, because we don't have the symbol available in e-mail.
We can also treat these names like variables, where we don't know what
specific functions we are calling f and g, yet we can say general
things about the relation of f and g, proving that something is true
for ANY functions, or at least for any functions of a certain type,
all at once. That is powerful!
Composition of functions takes two functions and makes a new one out
of them. The inverse of a function is a new function that has some
important properties; in fact, if you think of composition as a sort
of "multiplication" of two objects (which are whole functions), then
the inverse function is sort of a reciprocal. In fact, that's why we
use the notation we do for inverse functions. We've taken a familiar
idea from arithmetic and applied it to something far bigger, largely
just by having named functions.
Composition can be thought of as something like plumbing or electrical
wiring, where we buy parts off the shelf and connect them end-to-end
to make a new device or to wire a house or factory according to our
needs. We know how each pipe, wire, switch, etc. functions (pun
intended), and we know how they combine, so we can understand the
whole complicated system in terms of its parts. Without the function
concept, we couldn't do that in math.
The concept is most useful when you get to calculus, and find that the
derivative and the integral are operations on functions: given one
function f, you can make a new function f' out of it, that has certain
important properties. This moves math up one level from algebra. So
the concept of functions is essential for a good understanding of
calculus.
The concept of a function is also central to computer programming,
though the details are somewhat different there. Most of what a
programmer writes consists of 'functions' that do parts of the work
of the program. By designing functions that do little pieces, we can
string them together to do more complicated things without looking so
complicated. For example, the sqrt function I mentioned gets its name
from many computer programming languages that provide this and many
other built-in functions so we don't have to write them ourselves. By
making that just part of a more general concept of functions, we are
able to write our own functions, and then put those together to make
larger functions.
Here are several explanations of various aspects of these ideas:
Why use f(x)?
http://mathforum.org/library/drmath/view/54577.html
Are All Functions Equations?
http://mathforum.org/library/drmath/view/53273.html
Inverse Functions in Real Life
http://mathforum.org/library/drmath/view/54605.html
If you have any further questions, feel free to write back.
- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/